多模式广义Hopfield模型的亚稳定性
Metastability in the generalized Hopfield model with finitely many patterns
摘要：
本文继续了文[2]、[3]中对无序平均场模型亚稳行为的研究。本文考虑了具有多个独立模式$ξ_1,...,ξ_p$的广义Hopfield模型，其中模式具有独立同分布.组件并遵循$[-1,1]$上的离散分布。我们表明，亚稳态行为发生，并提供亚稳态退出时间和相应的能力上的尖锐的渐近性。我们应用潜在的理论方法开发的Bovier等人。在适当的序参数的空间，并使用离散拉普拉斯算子的分析，以获得容量的下界。此外，我们还考虑了多个鞍点具有相同速率函数值的可能性以及能量面在临界点附近退化的情况。
英文摘要：
This paper continues the study of metastable behaviour in disordered mean field models initiated in [2], [3]. We consider the generalized Hopfield model with finitely many independent patterns $ξ_1,...,ξ_p$ where the patterns have i.i.d. components and follow discrete distributions on $[-1,1]$. We show that metastable behaviour occurs and provide sharp asymptotics on metastable exit times and the corresponding capacities. We apply the potential theoretic approach developed by Bovier et al. in the space of appropriate order parameters and use an analysis of the discrete Laplacian to obtain lower bounds on capacities. Moreover, we include the possibility of multiple saddle points with the same value of the rate function and the case that the energy surface is degenerate around critical points.
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